Past exam of the quantum condensed matter course of the University of Cambridge / 2007 / Section A / A2

Long anisometric molecules cannot pack independently. In the Onsager excluded-volume mechanism, two slender rods of length $L$ and diameter $D$ exclude a relative-position volume approximately $2L^{2}D|\sin \gamma |$, where $\gamma$ is their mutual angle. Aligning rods reduces this excluded volume, increasing the translational entropy available at a fixed number density. It simultaneously reduces orientational entropy. At sufficiently high concentration the positional gain wins, producing a nematic liquid crystal even for purely repulsive rods.

The nematic director is apolar: $\mathbf{n}$ and $-\mathbf{n}$ describe the same orientation. The molecules retain liquid-like positional disorder, in contrast to a smectic phase with layered order. A typical dimensionless concentration controlling the slender-rod competition is $c{L}^{2}D$, although finite aspect ratio, flexibility and attractions change the transition. The mechanism illustrates that ordering can increase total entropy: the lost orientational freedom is compensated by less mutual obstruction.

At a micrometre scale, viscous forces normally dominate inertia. As an illustrative Newtonian estimate, $L=1\mu \mathrm{m}$, $U=10\mu \mathrm{m}{\mathrm{s}}^{-1}$, $\rho =10^3\mathrm{kg}{\mathrm{m}}^{-3}$ and $\eta =10^{-3}\mathrm{Pa}\mathrm{s}$ give a Reynolds number $\rho UL/\eta =10^{-5}$. The surrounding flow is approximately a Stokes flow, satisfying $-\nabla p+\eta {\nabla }^2\mathbf{u}=0$ and $\nabla \cdot \mathbf{u}=0$.

A body of radius $0.5\mu \mathrm{m}$ translated at that speed in an unbounded Newtonian liquid would need Stokes drag $6\pi \eta aU\simeq 9.4\times 10^{-14}\mathrm{N}$. It stops almost immediately when propulsion stops. The scallop theorem rules out propulsion by a single reciprocal stroke in this ideal regime; a rotating helix or a nonreciprocal cycle of shapes can propel it. Brownian motion also randomizes position and orientation: Stokes–Einstein relation scales as $a^{-1}$ and rotational diffusion as $a^{-3}$, so retaining heading becomes harder as size decreases.

The Newtonian calculation is a scale estimate, not a complete model of blood. Vessel walls alter drag, the imposed flow advects the body, and nearby red blood cells make the medium heterogeneous at this scale. Non-Newtonian fluid effects and deformable obstacles can matter; steering must be assessed against those effects rather than inertial coasting.

A lipid bilayer is fluid within its plane, so a bare equilibrated bilayer has no static in-plane shear modulus. Stretching its area is nevertheless expensive, whereas bending is much softer. The Helfrich membrane energy expresses this distinction throughwith membrane bending rigidity $\kappa$, mean curvature $H$, Gaussian curvature $K$, spontaneous curvature $C_0$ and membrane tension $\sigma$. Area dilation additionally costs approximately $K_A(\Delta A{)}^2/(2A_0)$. For a closed surface of fixed topology, the Gauss-Bonnet theorem makes the integrated Gaussian-curvature term constant.

These energies control vesicle shapes, budding and the ease of forming membrane tubes. Asymmetric leaflet composition or bound proteins can change the preferred curvature. Real cell membranes can also be coupled to a cytoskeleton, which supplies shear elasticity absent from a free lipid bilayer. Thus a red-cell shape cannot generally be inferred from the bare bilayer model alone.

Two useful measurements probe different terms. Optical tracking of thermal membrane flickering measures fluctuation amplitudes; for small height modes, a normalization with $h_q=A^{-1}\int h{e}^{-i\mathbf{q}\cdot \mathbf{x}}{d}^{2}x$ gives $⟨|h_q{|}^2⟩=k_{B}T/[A(\kappa q^4+\sigma q^2)]$. Micropipette aspiration controls pressure while measuring the membrane drawn into a narrow pipette, linking apparent area change to tension and, at larger strain, area dilation. The observables test the elastic coefficients rather than merely showing a membrane image.

In diffusion-controlled aggregation, particles meet through Brownian motion and stick with probability close to one on encounter. Diffusion-limited cluster aggregation allows already formed clusters to diffuse and join; diffusion-limited aggregation in its usual single-seed model instead adds diffusing individual particles to one growing cluster. First-contact sticking shields inner regions from later arrivals, favouring open branched structures. A reaction-limited cluster aggregation model has a small sticking probability and permits many failed encounters before attachment.

For two dilute spherical species, the Smoluchowski diffusion-controlled encounter rate is $k_{12}=4\pi (D_1+D_2)(a_1+a_2)$. In relative coordinates, solve the steady diffusion equation with an absorbing surface at $R=a_1+a_2$ and concentration $c_{\infty }$ at infinity: $c(r)=c_{\infty }(1-R/r)$. The inward diffusive flux is $4\pi (D_1+D_2)R{c}_{\infty }$, proving the rate coefficient. Interaction barriers and non-spherical aggregates require changes to this ideal encounter calculation.

Cluster growth can be tested by microscopy and light scattering. A mass fractal dimension $d_f$ relates particle number to cluster size through $N\propto R^{d_f}$; a scattering regime between the monomer and cluster length scales can reveal that scaling. The measured morphology depends on whether whole clusters move, on the sticking rule, and on possible bond rearrangement.